Again, we go beyond the book here.  (The good students will shout "HOORAY!"; the dumbos whom I couldn't care less about will shout "BOO!")  But, here goes...

Defining Unit Cells:

Unit cells are the basic structure of a crystal lattice.  Before we delve into their types, we need to look at how their axes are defined.  All unit cells are in essence "boxes"; the corner of the box is defined by three axes and three angles.  The axes are labelled a, b, and c and the angles are a, b, and g.  The angle between to a and b is g, the angle between a and c is b and, finally the angle between b and c is a .  (If you can get your Greek vs. Roman letters straight, it should be easy to see the the included angle is the "odd letter out" between the two Roman letters.)  This is shown by the following diagram.

Although not pretty, the diagram is informative.  (Provided you can read my writing.)
 

Primitive lattices are the simplest possible crystal lattices.  With them, the particles forming the lattice are just at the corners.  The primitive lattices define the shape of the unit cell and are listed in the next figure.  Cubic, tetragonal, and orthorhombic cells have all angles right angles.  Cubic cells of course have all sides equal; tetragonal cells have two sides equal (forming identical squares at opposite ends); orthorhombic cells have all faces as rectangles.  Trigonal cells have all sides equal but the angles not right angles; these are basically "squashed cubes"; these cells are also sometimes called orthorhombic.  The three remaining primitive cells have successively less symmetry.  Hexagonal cells are a special case since they are prisms with two angles right angles and the third exactly 120^o (carved out of a hexagon).  This is an especially useful symmetry in certain cases.  Finally, monoclinic cells have two right angles and a third angle which is neither 90 nor 120 degrees; all three defining sides are unequal.  Finally, triclinic unit cells have no 90 degree angles and all three defining sides unequal.  We show these in the following picture.

 

The seven primitive systems are the only ones possible at the simplest level.  However, there are seven more lattices possible when we allow particles to be on the ends, faces, or interior of a unit cell.  The 19^th century scientist, Bravais showed that, in total there are 14 possible crystal systems if one includes these extra locations.  For instance, in the book you see the three simple cubic unit cells:  simple cubic, face-centered cubic, and body-centered cubic.  The complete list of Bravais lattices is shown in the next figure.

We can count the types of each Bravais lattice belong to each primitive (simple) system and do in the table below.
 
 

Primitive (Simple) vs. Bravais Crystal Lattices

Primitive (Simple) Type	Bravais Types	Number*
Triclinic	Simple	1
Monoclinic	Simple Side-Centered	2
Orthorhombic	Simple End-Centered Face-Centered Body-Centered	4
Hexagonal	Simple	1
Rhombohedral (Trigonal)	Simple	1
Tetraganol	Simple Body-Centered	2
Cubic	Simple Body-Centered Face-Centered	3

*There are 14 total Bravais lattices.

 

What Bravais did was probably a matter of trial and error with some math thrown in.  Later on, when a branch of mathematics called group theory was discovered, it was found that symmetries within crystal lattices could be further related to molecular symmetry to generate what are called Schoenflies lattices.  We won't go into details here but there are 32 crystallographic point groups.  These can be combined with the Bravais lattices to produce the 230 possible space groups.  These correspond to the exactly 230 lattices possible in total.  (This strange number arises since some combinations of point groups and Bravais lattices are redundant.)  You might wonder how--or even why--these were worked out.  It was simple, Schoenflies was a prisoner of war in a concentration camp and he worked these out to keep his sanity!

Needless to say, we won't show all these space groups.  However, if you are really curious, get a very good book on group theory and crystallography and do a few days' hard study!